Drago Bokal, Gašper Fijavž, David R. Wood
The "minor crossing number" of a graph $G$ is the minimum crossing number of a graph that contains $G$ as a minor. It is proved that for every graph $H$ there is a constant $c$, such that every graph $G$ with no $H$-minor has minor crossing number at most $c|V(G)|$.
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Every gammoid is a minor of an excluded minor for the class of gammoids.
We find the excluded minors for the minor-closed class of lattice path polymatroids as a subclass of the minor-closed class of Boolean polymatroids. Like lattice path matroids and Boolean polymatroids, there are infinitely many excluded minors, but they fall into a small number of easily-described types.
Excluded minors of interval positroids that are paving matroids
本文研究区间正拟阵的排除子式，证明铺砌情形可通过松弛相关超平面归约为三个基本族，并据此完成非正拟阵情形的分类，同时给出铺砌正拟阵情形的判定准则。
The excluded minors of the class of spike minors
本文解决了Mayhew等人提出的问题，证明了尖峰图及其子式类具有有限排除子式集合并给出了完整描述。
We present infinite sequences of excluded minors for both the class of lifted-graphic matroids and the class of frame matroids.
We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if $\mathbb{F}_1$-representability is not equivalent to $\mathbb{F}_2$-representability, then each $\mathbb{F}_1$-representable matroid is a minor of a $\mathbb{F}_2$-representable excluded minor for $\mathbb{F}_1$-representability.
The excluded minors for GF(5)-representable matroids on ten elements
针对GF(5)可表示拟阵的排除子拟阵问题，本文枚举得出10个元素的该类拟阵有2128个排除子拟阵，还同时得到了至多10个元素的ℍ₂、ℍ₃、ℍ₄可表示拟阵及至多13个元素的ℍ₅可表示拟阵的排除子拟阵。
We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid.
In this paper, we first introduce the notions of checkerboard colourable minors for ribbon graphs motivated by the Eulerian ribbon graph minors, and two kinds of bipartite minors for ribbon graphs, one of which is the dual of the checkerboard colourable minors and the other is motivated by the bipartite minors of abstract graphs. Then we give an excluded minor characterization of the class of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.
Lattice path matroids: the excluded minors
本文通过确定被排除的未成年，完整刻画了格路径拟阵的未成年封闭类。
Computing excluded minors for classes of matroids representable over partial fields
本文实现了一种计算机搜索方法，用于寻找可在部分域上表示的拟阵类的“小”排除子式，枚举了二元和2-正则拟阵类至多15个元素的排除子式，并对两者的排除子式提出了猜想与证明。
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic flats. We confirm a conjecture of Joswig and Schröter by proving an excluded-minor characterisation of the class of split matroids.
If EE is a set of matroids, then ex(EE) denotes the set of matroids that have no minor isomorphic to a member of EE. If EE' is a subset of EE, we say that EE' is /superfluous/ if ex(EE - EE') - ex(EE) contains only finitely many 3-connected matroids. We characterize the superfluous subsets of six well-known collections of excluded minors.
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both the class of lifted-graphic matroids and the class of frame matroids, each of which generalises the class of graphic matroids. In this paper, we prove that the matroids $U_{3,7}$ and $U_{4,7}$ are the only $9$-connected excluded minors for the class of quasi-graphic matroids.
There are only a finite number of excluded minors for the class of bicircular matroids
本文证明了双圈拟阵类仅有有限个排除子式，核心方法是利用偏置图和拟图拟阵表示，证明秩至少为十的排除子式必为拟图拟阵并包含已知小排除子式，从而得出结论。
The excluded minors for the intersection of bicircular and lattice path matroids
本文针对双圈拟阵与格路拟阵的交集类别，确立了其完整的排除子式列表，从而解决了一个最近提出的开放问题。
Let $G$ be a graph with $n$ vertices, with independence number $α$, and with with no $K_{t+1}$-minor for some $t\geq5$. It is proved that $(2α-1)(2t-5)\geq2n-5$.
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $\mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $\mathcal{M}$ has only a finite number of excluded minors of rank $r$.
Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded relative to $N$, or, up to replacing $M$ by a $Δ$-$Y$-equivalent excluded minor, we can choose a pair of elements $\{a,b\}$ such that either $M\backslash \{a,b\}$ is $N$-fragile, or $M^* \backslash \{a,b\}$ is $N^*$-fragile.
The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors that essentially says that such graphs can be decomposed into trees of graphs of bounded local tree-width. As an application of this theorem, we show that a number of combinatorial optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set have a polynomial time approximation scheme when restricted…
The excluded minors for $k$-polymatroids with binary $k$-natural matroids
本文研究了具有二元 $k$-自然拟阵的 $k$-多拟阵类的排除子式问题，确定了当 $k \geq 3$ 时该类的排除子式集合是有限的，其中 $\widetilde{\mathcal{C}}'_3$ 有 $12$ 个，$k > 3$ 时有 $k+7$ 个。
A graph is nearly embedded in a surface if it consists of graph $G_0$ that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly embedded graph, many rows of which intersect the embedded subgraph $G_0$ of the near-embedding, contains a large subwall that is planarly embedded within $G_0$. This result provides some hidden details needed for a strong version of the Robertson and Seymour's excluded minor theorem as presented in [K. Kawarabayashi, B. Mohar, Some recent progress and applicat…
An infinite family of excluded minors for strong base-orderability
本文通过推广Ingleton的结果与例子，构造了无限多个基可序拟阵作为强基可序性的排除子式，并引入k-基可序性刻画其层次结构。
The excluded minors for 2- and 3-regular matroids
本文刻画了2-正则拟阵类的排除子式，并证明3-正则拟阵类与Hydra-5部分域可表示类一致，其排除子式至多含15个元素。
Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid $L(G)$ are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. For each cycle $C$ of $G$, since $L(G)/C$ is graphic and most graphic matroids are not bicircular lift, the class of bicircular lift matroids is not minor-closed. In this paper, we prove that the class of matroids that are graphic or bicircular lift has a finite list of excluded minors.
Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus $k$ or for embeddings into nonorientable surface of genus $k$ are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained.
Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bo…
In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdière parameter at most $m$. We also characterize graphs of maximum spectral radius with no $H$ as a minor when $H$ is either $K_r$ or $K_{s,t}$. Interestingly, the extremal graphs match those which maximize the number of edges over all graphs with no $H$ as a minor when $r$ and $s$ are small, but not when they are larger.
We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with no $K_p$-minor satisfies $|E(G)|\leq (p-2)|V(G)| - (p-2)^2$.
Let $V_{8}+e$ denote the unique graph obtained from the Wagner graph, also known as $V_{8}$, by adding an edge between two vertices of distance 3 on the Hamilton cycle, which is exactly a split of a minor of the Petersen graph. A complete characterization of all internally 4-connected graphs with no $V_{8}$ minor is given in [J. Maharry and N. Robertson, The structure of graphs not topologically containing the Wagner graph, J. Combin. Theory Ser. B 121 (2016) 398-420]. In this paper we characterize all internally 4-connected graphs with no $V_{8}+e$ minor.
Let ${\rm ex \,} {\mathcal B}$ be a minor-closed class of graphs with a set ${\mathcal B}$ of minimal excluded minors. We study (a) the asymptotic number of graphs without $k+1$ disjoint minors in ${\mathcal B}$ and (b) the properties of a uniformly random graph drawn from all such graphs on vertices $\{1,\dots,n\}$. We present new results in the case when ${\rm ex \,} {\mathcal B}$ contains arbitrarily large fans for a general (good enough) set of forbidden minors ${\mathcal B}$. A particular case where our results hold is ${\mathcal B} = \{K_4\}$. For any fixed $k = 1, 2, \dots$ we derive pr…
In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.
In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization.
The essential bound of a polymatroid and its applications to excluded minor problems
本文通过引入多拟阵的有界性概念，研究k-多拟阵的排除子式问题，并证明当k-自然拟阵属于排除U_{2,b}及其对偶的闭子式类时，排除子式列表有限。
The excluded minors for embeddability into a compact surface
该研究确定了刻画可嵌入某紧曲面的可数图类的禁用子式，推进了图论中曲面嵌入相关的子式理论研究。
In this paper we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterisations of minor closed families of ribbon graphs. Our main result is an excluded minor characterisation of the family of ribbon graphs that represent knot and link diagrams.
As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded genus. Recently, Grohe and Marx proved a similar result for graphs not containing $H$ as a topological minor. They showed that every graph which does not contain $H$ as a topological minor has a tree-decomposition in which every torso is either almost embeddable in a surface of bounded genus, or has a bounded number of vertices of high degree. We give a short pro…
Almost generalized uniform matroids and excluded minors
本文通过Tutte多项式系数刻画了几乎广义均匀拟阵，并构造了其排除子式的无限族，推广了已知结果。
An infinite graph G is minor excluded if there is a finite graph that is not a minor of G. We prove that minor excluded graphs have finite Assouad-Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators; (4) minor exclusion is preserved under free products; and (5) …
Classes of binary matroids with small lists of excluded induced minors
该研究在前期拟阵诱导子式相关刻画工作的基础上，确定了不含指定二元拟阵对作为诱导子式的拟阵类，并证明了用于刻画不含$M(K_4)$诱导子式拟阵类的结构引理。
An excluded minor theorem for the 6-wheel
本文研究6轮图的极小图特性，通过分析平面图与非平面图的结构，完成对不含6轮图作为子图的图的完整分类。
The excluded minors for three classes of 2-polymatroids having special types of natural matroids
本文针对自然拟阵分别属于二元拟阵类、不含M(K4)-子式的拟阵类以及串并联网络圈拟阵连通分支类的三类2-多拟阵，求出了它们的排除子式，并引入k-自然拟阵证明这些类在2-对偶下封闭。
The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are representable over k. We expect that the method we use to prove this will make it possible to lift many theorems about finite matroids representable over a finite field to theorems about tame thin sums matroids over these fields. We give three examples of this: various characterisations of binary tame …
Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either G\v is planar for some vertex v, or G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.
At the core of the Robertson-Seymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some particular aspects needed at a corresponding stage of the proof of the main result of their theory, the graph minor theorem. We prove a new version of this structure theorem: one that seeks to combine maximum applicability with a minimum of technical ado, and which might serve as a canonical version for fu…
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has l…
The excluded 3-minors for vf-safe delta-matroids
本文通过寻找最小障碍即排除3-minors来刻画vf-safe delta-matroids和ribbon-graphic delta-matroids，并解释了binary delta-matroids中3-minors与图vertex minors的等价性。
Minor-closed classes of binary functions
本文研究二元函数的小闭类，通过排除小元刻画具有良定义小元及关联秩函数的类，并推广Tutte定理至二元函数空间。
We prove that if $G$ is a graph with an minimal edge cut $F$ of size three and $G_1$, $G_2$ are the two (augmented) components of $G-F$, then the crossing number of $G$ is equal to the sum of crossing numbers of $G_1$ and $G_2$. Combining with known results, this implies that crossing number is additive over edge-cuts of size $d$ for $d\in\{0, 1, 2, 3\}$, whereas there are counterexamples for every $d\ge 4$. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces…
The Excluded Vertex-Minors and Pivot-Minors for Rank-Width at Most Two
该研究确定秩宽至多为二的图类的排除顶点子式和排除枢轴子式，通过反转素图单顶点约化定理，利用规范键确定等价类，借助计算机辅助计算得出具体数量，完善了相关列表。
A characterization of positroids, with applications to amalgams and excluded minors
本文给出拟阵胚的一个新刻画，并利用它证明两个拟阵胚在交集为独立克隆集时其自由融合仍为拟阵胚，同时构造出拟阵胚类的多参数无限族排除子式。
The excluded minors for $\mathsf{Z}_{3}$-gainable and regular biased graphs
通过发展部分群理论，证明偏斜图在Z3上可增益当且仅当不含特定子式，并推导出Gerards定理。
A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k>4) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the …
Excluded conformal minors of Birkhoff-von Neumann graphs with equal global forcing number and maximum anti-forcing number
本文针对二分图和Birkhoff-von Neumann图，通过耳分解定理等方法，刻画了全局强制数等于最大反强制数的强一致可匹配图的被排除共形子式与结构特征。
Excluded minors are almost fragile II: essential elements
本文证明在温和假设下，被排除的 minor 的拟阵 $M\backslash a,b$ 离一个具有至少 $r(M)-2$ 个 $N$-本质元素的 minor 仅一个元素的距离，延续前篇关于脆弱性的研究。
We give an excluded minor characterisation of the class of ribbon graphs that admit partial duals of Euler genus at most one.
We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a compactification of the space of countable (connected) rooted graphs, and use it to generalise the notion of Benjamini-Schramm convergence in order to allow for vertices of infinite degree in the limit object.
The Excluded Tree Minor Theorem Revisited
本文证明了排除树子式定理的定性强化版本，指出半径是正确参数且 $2h-1$ 是最佳路径宽度界限。
A quasi-polynomial bound for the minimal excluded minors for a surface
本文将曲面极小排除子式的阶上界从双重指数改进为欧拉亏格的拟多项式，关键突破是将禁止结构的特征尺寸从 $O(g)$ 降至 $O(\log g)$，并改进了树宽界。
Product structure of graphs with an excluded minor
本文证明无 $K_t$ 或 $K_{s,t}$ 子式的图可嵌入树宽有界的树状图与完全图的乘积，并给出最优树宽和顶点数相关界，推广了分隔符定理和平面图产品结构定理。
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